On 4 Jun., 04:21, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > In article <5b9b7d5b-11d3-4125-8052-ad45e6e7a...@x5g2000yqk.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> writes: > > On 3 Jun., 04:25, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > > > There is no difference between finite unions and infinite > > > unions. > > > > There should not be, but there is --- at least if ZF is taken to be > > true. > > You are again wrongly interpreting things. There is no question about ZF > being true or not. ZF is only one of the many possible theories.
It is an impossible theory because finished infinity is impossible. > > Mathematics is not a science that tries to find truth,
There is a mathematics, namely basic arithmetic of integers, that is right. And there are some logical rules that are right. And from these foundations some theorems can be obtained that are right. There is no need and no space for any axiom at all. There is no need , in particular, for "a set N" with some "binary relations". It is impossible, to change that stuff. And Hilberts famous model including 1 + 1 = 0 does nothing but express the display of the last wheel of a binary calculator (or Leibniz's calculator that all time long had problems with the digit transfer).
> it is a science where > it is determined what a given set of axioms (i.e. presupposed valid statements) > leads to.
A given set of axioms is sometimes impossible. This is so with the axiom of infinity that Zermelo created, misunderstanding Dedekind's idea of potential infinity.
> > The union of FISONs (of natural numbers or of other finite linear > > sets) is the last FISON. > > ZF does agree, when there is a last FISON. Your problem is that the axiom > of infinity states that here is no last FISON of all FISONs.
That is not my problem. For potentially infinite sets it is not a problem at all. For a completely existing set ZF is wrong. But that is usually veiled by applying potential infinity whenever the problem occurs.